I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, invariant quantities are described as being those physical quantities associated with a point in space e.g. temperature at point P. Thus, when changing coordinate system, the value associated with the point is unchanged: you may represent the point by a different set of coordinates, but on locating the point in the new coordinates, the temperature value will still be the same. Here comes my struggle: the archetypical example of a covariant quantity is the gradient vector. Is it not the case that we require it to transform between coordinate systems in a way that preserves the vector associated with each point? Thus, in a way, the gradient vector is an invariant quantity? Why then are many derivations in my course driven by the fact that physical quantities like Action, Lagrangians etc should be "Lorentz invariant"?

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When we say that a vector is invariant, we mean that the vector does not change under a change of coordinate systems. This does not mean that the components of this vector are invariant, however. It is to the components of Tensors that the terms contra-variant and co-variant apply.

Consider a vector $v=(1,0,0)$, that is, a vector of unit length pointing along the $x$-axis. It has components $v_x=1$, $v_y=0$, and $v_z=0$.

Now apply a coordinate transformation, such that we rotate our point of view by 90 degrees about the $z$-axis. In our new coordinate system, our vector has a new representation: $v'=(0,-1,0)$. While the vector itself is unchanged, the vector components are not. In this new frame, $v'_x=0$, $v'_y=-1$, and $v'_z=0$.